The geometric resistivity correction factor for several geometrical samples

Transcription

1 Vol. 36, No. 8 Journal of Semiconductor Augut 05 The geometric reitivity correction factor for everal geometrical ample ; ; Serdar Yilmaz Merin Univerity Science and Art Faculty Phyic Department, Merin, TR-33343, Turkey Merin Univerity Advanced Technology Education, Reearch and Application Center, Merin, TR-33343, Turkey Abtract: Thi paper review the geometric reitivity correction factor of the 4-point probe DC electrical conductivity meaurement method uing everal geometrical ample. During the review of the literature, only the article that include the effect of geometry on reitivity calculation were conidered. Combination of equation ued for variou geometrie were alo given. Mathematical equation were given in the text without detail. Expreion for the mot commonly ued geometrie were preented in a table for eay reference. Key word: emiconductor; four point probe; conductivity meaurement; reitivity correction factor DO: 0.088/ /36/8/0800 EEACC: 50. ntroduction There i a functional relationhip between the teted reitivity () of a material and it geometric hape. Therefore, a geometric reitivity correction factor (G/ i ued for calculating the reitivity with R D V = equation. Thi correction factor change with the thickne of the ample (t/, geometric dimenion, the area of the urface (A/, the poition and the array of the probe on the ample Œ. The reitivity i given by Equation () a a general Equation Œ. The conductivity meaurement i affected by everal factor uch a polarization, impurity, ample geometry, cable reitance, temperature and humidity Œ3. According to the ample geometry and meaurement technique one or more factor are dominant. Thu the reitivity meaurement technique i decided with the condition. D V G; () where i the reitivity in cm. The total reitance (R total / value can be meaured with the tandard -probe meaurement technique and thi parameter i equal to the ummation of the reitance of the ample (R ample /, the wire (R wire /, the probe (R probe / and the conductive pate (R pate /. The pate i located between probe and ample. Therefore, calculated reitivity with thi method i higher than the real reitivity of the ample. The difference between the real and calculated reitivity of a ample become more important when the reitivity of the probe are higher than the ample reitivity. The reitivity of the ample (R ample / can be meaured uing the 4-probe technique. n thi meaurement technique, probe numbered and 4 are ued for meauring the current on the urface ( /, probe numbered and 3 are ued for meauring the potential difference between any two point on the urface (Figure ). Due to the lack of a current between probe and 3, no voltage drop in thi line and probe. n thi meaurement technique, the reitivity i given by Equation (). Figure. The meaurement with random four probe on urface. D V 3 4 G: () The meaurement et-up that ha in-line probe arrangement i a practical application (Figure ). n thi meaurement et-up, i the ditance between probe in cm and D 3 D 34 D in general. n thi tatement the determining and calculating of the G factor i eaier. Correponding author. yilmaz@merin.edu.tr Received 7 January 05, revied manucript received March 05 Figure. Meaurement et-up with four in-line array probe. 05 Chinee ntitute of Electronic 0800-

2 Table. t and relation commonly ued for an infinite extent. t t nfinite thick Thick Thin (heet) nfinite thin (thin film) t > 5 t > t < t < 0. Reitivity meaurement of the emiconductor The current denity (J / i directly proportional to the electrical field magnitude (E/ for the conductive material (Equation (3)) Œ4. D E J D E =A : (3) The work (W / applied by E to the charge which have flow through L way and between any randomly taken a and b point i given imply by Equation (4) Œ5; 6. According to thi expreion, for determination of the ample reitivity both electrical reitance and geometric extent hould be known (Equation (4)). Figure 3. Electrical current on the conductive layer. W D U D.U b U a / D q Z b a EdL; (4) V ab D V b V a D U b U a q D E J D V ab=l =A D D V ab Z b a EdL D EL; (5) A L D V ab G: (6) Many emiconductor material have a high reitivity. Becaue the reitance of emiconductor can rapidly decreae with temperature, the reitance of probe have become very important. The pecially deigned probe which have fine contact between the probe and the ample are ued to meaure reitivity. Thee probe have, in general, harp point and a mall body. When probe have a mall body, they have weaknee in regard to mechanical force and high reitance. Thu in practice very hard material are ued to make probe uch a omium or tungten alloy. Therefore, the reitance of probe increae when the reitivity i meaured at high temperature. For thi reaon, the primary method ued for reitivity meaurement i the 4-point probe technique. However, in application, the reearcher hould note that in order for the metal probe to not caue damage to the thin-film ample, the preure of probe on the ample hould be mall. Nonethele, advere effect of the charge tranfer diffuion are minimized Œ7. 3. Geometrical reitivity correction factor The tarting point of the geometrical factor calculation i mot commonly the formulation of G for ample with infinite extent and thickne. The expreion of the ample a thin or thick depend on the ratio between the thickne of the ample (t/ and the pacing of the probe (/. Practically uing the ratio of and t for a lice of infinite extent i given in Table. n thi table t and i the ample thickne and the probe pacing in cm, repectively. Figure 4. The ample with t thickne. For a emi-infinite ample, the bulk reitivity i given by Equation (7) Œ; 8; 9. The emi-infinite ample i a plane that ha an infiniteimal thickne and urface area. The probe are located on thi plane. bulk D V G D V : (7) Although infinite extenion, if the ample i a thin layer, the reitivity i computed a Equation (8) Œ; ; 0 5. D V G D V t: (8) ln f the ample i an infinite heet, t i negligible. Thu, the heet reitivity i calculated from Equation (9) Œ;. 0 D heet D V G D V ln : (9) n Equation (9), heet i reitivity of a heet or an infinite thin ample urface ( urface /. n pite of the calculated unit of reitivity in Equation (9) being, t i negligible but it unit i in reitivity equation Œ. f the ample ha a finite thickne and finite extent, new geometric correction are added to Equation (7) a a factor. 3.. Semi-infinite volume f the ample i a emi-infinite volume, the reitivity expreion doe not include the parameter of t and urface area 0800-

3 Figure 5. Y point away from ingle polar X point. Figure 7. The two different point away from current dipole with r ditance. Figure 6. The Y point that r and r ditance from electrical dipole. for any hape. Ditance between probe i, 3, 34, repectively. Firtly, the current flow from X point to ample a a ingle polar (Figure 5). The charge are carried to Y point from thi point. r i the ditance between X and Y. The potential of Y i obtained a Equation (0) via calculating from Equation (5). V Y D V D Er D q r : (0) Uhlir (955) had given Equation (0) for infinite urface extent a below. D AE D r q r D q : () n thi expreion the A i the urface area of a emi-phere with infinite mall r radiu. The center of a emi-phere i the contact point between the ample and the current probe. The V voltage for r ditance from X point (Equation ()) i obtained with combining Equation (0) and () Œ; 8; ; 6. V D r : () n thi ituation, the voltage i expreed a Equation (3) for the dipole current ource a een in Figure 6. f the meaurement i performed on four different point and ample a een in Figure 7, voltage drop between probe and 3 i decribed a Equation (4). V Y D r r D V 3 D V 3 V D C r 3 r 34 r r 4 ; r r (3) : (4) f the arrangement of electrical contact i in-line array on the ample (Figure ), potential difference i given a Equation (5) and the reitivity i obtained a Equation (6) Œ8; 6. n thee equation, only the parameter i finite and meaurable. The thickne or the urface geometry i approximately infinite. For thi reaon, the G only depend on the. f ditance of the probe are given a equal ( D 3 D 34 D /, according to Equation (5) the correction factor i computed a Equation (7) for the emi-infinite volume Œ8; 6. f the meaurement et-up i organized for the D = D 0.59 m, reitivity value i calculated a D V=. V 3 D V D C 3 C 34 C D V C D 34 C 3 3 C 34 ; C 3 34 (5) V G; (6) G D F 0./ D : (7) 3.. nfinite urface area and t thickne Figure 4 belong to a lice of infinite extent and meaurable t thickne. The G factor in reitivity expreion hould include parameter of the thickne for thi ample. t uppoe to t (t > = in practice) the ample i thick ele (t < = in practice) the ample i thin (Table ) Thick ample The reitivity for the thick ample in t > = i expreed a Equation (8) Œ; ;. n thi expreion F.t=/ i the extra geometrical correction factor for the finite t thickne ample and i expreed a Equation (0) Œ7. Thi equation i more ueful and imple than the one decribed by Uhlir (955) (Equation (3)). D V F 0./ F.t=/; (8) G D F.t=/; (9) F.t=/ D t= ln inh.t=/ inh.t=/ : (0) F.t=/! while t! (Table ). With reference to thi approach, extra geometrical correction factor can be given a F.t=/ D for t= > 5 Œ5. n thi tatement, the ample thickne can be accepted a emi-infinite for the t > 5 thickne and geometrical factor i tranform to Equation (7)

4 Table. F.t=/ factor value veru t= ratio. t= F.t=/ Table 3. The effect of the thickne. t= inh.t=/ inh.t=/ Figure 8. B reference point on the ample with emi-infinite extenion and t thickne Thin ample inh.t=/ Š can be acceptable for the finite and thin meaurable lice and t < = thickne (Table 3) and the function inh.t=/ of F.t=/ tranform to Equation () Œ6. F.t=/ Š t= ln : () A can be een in Table 3 and Equation (), if t < =, ha great value compared with the thickne of the ample. n thi cae the reitivity expreion for the thin ample which ha three-dimenional geometry, infinitive extenion and finite but mall t thickne i a in Equation () Œ; ; 5; 0 6. G D F 0./ F.t=/ D t: () ln Uhlir (955) obtained the ame reult uing a different method of calculation Œ6. Thi tudy i the firt in the literature that contain the geometric factor equation taking into account thickne. Becaue the current flow through a urface t unit thick in an infinitely large volume, the volume of the ample i not conidered to be infinite. n thi cae the reitivity expreion hould contain the thickne term and Uhlir expreed uch a Equation (3) Œ6. n thi equation CD (correction divior) i an additional correction factor which i derived for the finite thickne and /CD D F Uhlir.=t/. D V CD ; (3) G D CD D F 0./ F Uhlir.=t/: (4) Uhlir, olution of the F Uhlir.=t/ function for the t < = given a Š ln Œ6. f thi expreion i ued in Equation (4), Equation () i obtained t again nfiniteimal thin ample The reitivity correction factor for an infiniteimally thin.t Š 0/ and infinite extenion wa firt reported by Smit (958) and Vaughan (96) Œ; ; 4. The beginning expreion wa achieved for the reitivity equation. Figure 8 i a ample which ha t thickne and emi-infinite extenion. A and B point are arbitrary location on thi ample. The A point i ingle polar current ource and the B i any under the tet Figure 9. Randomly elected the B point that i r ditance from current dipole A and C point. point. Alo r i the ditance between point A and B. n thi tatement the potential difference between point A and B i decribed a Equation (5) Œ5; 6. Z Z V D Edr D J dr D ln r; (5) t V D V V 0 D ln r: (6) f thickne i infiniteimal.t Š 0/, the potential difference i given a Equation (6) for a point on the lice Œ; 4. n thi equation V i the meaured potential and V 0 i the reference potential. The potential difference on the randomly elected ingle B point location for the dipole current ource which i on the t Š 0 lice i given a Equation (7) (Figure 9). On the other hand the potential difference on the randomly elected two point location on the t Š 0 lice i expreed a Equation (8) (Figure 0). V B D ln r ; (7) r V D V 3 V D ln r 3 ln r D r 34 r 4 ln r 3r 4 : r 34 r (8) f the probe ditance are equal. D 3 D 34 D / and four-probe point are in-line array (Figure ), the voltage difference between probe and 3 and the geometric contribution are obtained a Equation (9) and (30), repectively. V D V 3 V D V D ln D ln ; (9)

5 Table 4. Computed F value for different d= ratio. d= F.d=/ tion (3)) Œ;. V D " # ln C ln.d=/ C 3.d=/ : (3) 3 Figure 0. Randomly elected two point ( and ) that are r ditance from current dipole and 4 point. n thi cae the reitivity for the circular, t < = and with d radiu ample i expreed a Equation (3) Œ; ; 3. 8 " # 9 D V < : ln C ln.d=/ C 3 =.d=/ 3 ; D V " # C ln ln ln.d=/ C 3.d=/ : (3) 3 Then the reitivity equation and additional correction factor i given by Equation (33) and (34) for the circular ample, repectively Œ; 0; ; 3. The olution of F.d=/ function are reported by Smit (958) and Swartzendruber (964) and reult are given in Table 4 Œ;. A een in Table 4, if.d=/!, F.d=/!. n thi cae the G expreion tranform to Equation (30) for the.d=/ > 40 geometry. Equation (30) belong to finitely and emi-infinitive lice.d! /. D V G D V h i ln F.d=/ ; (33) Figure. nfiniteimal thin and circular urface Circular ample G D ln : (30) n thi part, a circular ample i oberved for a lice of finite extenion, d radiu and finitely or very thin thickne. The probe are in-line arrangement, centered on the circular urface and equal ditance (/ on the ample. Again, the G change depending on the t thickne of ample for the circular ample with finitely urface area. Thee different condition on thickne hould be invetigated repectively nfiniteimal thin circular ample Figure indicate the circular ample with finite urface area and infiniteimal t thickne. So thickne i ignored. n thi ytem, ditance between probe (/ and tatement of the probe on the ample are very important. A ratio function of circle radiu to probe pacing contribute to the potential (Equa- F.d=/ D " # C ln ln.d=/ C 3.d=/ : (34) 3 Vaughan (96) obtained the ame equation for the circular ample uing different mathematical method. The reitivity i expreed below for an in-line arrangement of fourprobe, auming an infiniteimally thin (heet) and a circular geometry Œ4. D V G D V ln 4L : (35) Here the term L i additional correction factor that i obtained by Vaughan (96). Thi correction factor include the t thickne, r radiu and a function that include coordinate of ditance between the center of in-line arrangement probe and current dipole. According to the calculation in thi publication, the reult of L equation i equal to L D for t=r D 0. n thi intance lim L D namely the ample i infinitely thin. So the t!0 urface reitivity i obtained again a Equation (30) for the two dimenion geometry and infinitely thin (thin film) ample. D V ln 4 D V ln : (36)

6 (d/) and ditance between center of the in-line array probe (Figure 3) and circle, repectively. n Equation (39), placement of the probe on the ample are taken into account during the calculation of geometric factor. f the probe are placed on the circle center ( D 0), the equation of i computed a Equation (40). n thi cae, if the equation i ued in reitivity expreion (Equation (38)), Equation (34) i re-obtained. Figure. Thin circular ample with t thickne. Figure 3. Placement of probe on circular ample Thin circular ample The G function of a circular ample which ha finite extenion and thin t thickne i given a below (Equation (37)) Œ; 0 3; 8. D V G D V ŒF 0./ F.t=/ F.d=/ D V ln t F.d=/: (37) Equation (37) i the geometric factor that include the effect of finitely t thickne (Equation ()), ditance of probe and urface dimenion limited with d diameter (F.d=//. n here, mean of the thin thickne i t= and in practice t= < 0:5. Swartzendruber (964) reported the F.d=/ function in general form Œ3. Swartzendruber (964) and Yamahita (989) explained the reitivity expreion for a circular ample a Equation (38). D V ln t C D 0. C / D 0 F.d=/ : (38) n Equation (38), the dimenionle magnitude i reported by Swartzendruber (964) and Yamahita (989) performed by different calculation but with the ame reult a Equation (39) Œ3; 8. n Equation (39), R and are radiu D ln R C 3 R R R R R R C 3 R ln ; R D ln ln R 3 R R 3 d d 3 d d R C d d R R C 3 R 3 d 3 d (39) " # D ln ln C 3.=d/ 3.=d/ : (40) The mot ueful ide of the equation, it include the effect of the probe array on the ample for all ample material hape. A een in Table 4, becaue lim F.d=/ D, the geometric factor i given a F.d=/ Š for d= > 40 geometry. d! n thi intance the G factor i re-obtained a Equation () for finite extenion and thickne Thick circular ample f t thickne i not mall.t= > 0:5/, the volume reitivity expreion i given a Equation (4) Œ47. Thi expreion include Equation (7), (0) and (34) function. D V F 0./ F.t=/ F.d=/ D V G: (4) 3.4. Rectangular ample Yamahita (987) invetigated the effect of thickne of the probe and geometry of the rectangular prim ample on the geometric factor. The rectangular prim ha mall t thickne (Figure 4). The contact point between current probe numbered and 4 ( and 4 / and the ample are the equivalent center of the current ditribution in the current probe. The diplacement ı i the ditance between the center of the probe and the current ditribution center. The probe numbered and 3 ( and 3 / are voltage probe. The V 3 i given by Equation (4) Œ. V 3 D 4 t ln. C ı/. C 3 ı/ : (4). ı/. 3 ı/ Equation (4) reitivity equation include t and parameter. n thi equation the diplacement ı (proximity effect) i given by ı D L.L 4R / =. L i the pacing between the and 4 probe where L D C C 3 and R i radiu of the probe tip. f the equivalent current ditribution

7 Table 5. The Geometric Factor equation for the commonly ued geometrie. Thickne Surface area Geometric factor equation (G/.t > 5/ G D F 0./ D C 3 C C 3 G D F 0./ D. D 3 D 34 D / t Thick.t > =/ G D F 0./ F.t=/.t > 5/ ) F.t=/ D Thin G D ln t.t < =/ nf. thin G D ln.t < =0/ Circular Thick.t > =/ G D F 0./ F.t=/ F.d=/.t > 5/ ) F.t=/ D Thin G D ln t F.d=/.t < =/ nf. thin G D ln F.d=/.t < =0/ Gegenwarth (968) reported the correction factor for a cylindrical emiconductor that ha an infinitely long radiu R Œ9. The reitivity function include the current denity, the area of contact of the electrode, the probe pacing, and a modified Beel function. The reitivity equation for the infinitely long bulk emiconductor (Figure 5) i decribed a Equation (44) Œ9. n thi equation K i an expreion in the potential equation which depend on the value and the poition of the probe. Figure 4. Rectangular prim ample with t thickne and probe with R radiu. D V R K D V G: (44) f Equation (44) i ued for the R meaurement etup, the G equation i computed again a Equation (7) Œ9. n thi computing, R ha to go to infinity. Thi reult i the ame with the emi-infinitive volume tatement and independent of the thickne. Figure 5. Cylindrical ample with R radiu. center i centered on the probe or the radiu of the probe are zero (pinpoint probe) or D D 3 D (o L D 3/, proximity effect become ı D 0. n thi tatement Equation (4) become Equation (43) and the G factor i obtained a Equation (). V 3 D 4 ln : (43) t n thi expreion, the G increae rapidly due to the experimental reult for the t > 0:5 mm and higher value. The contribution of t ha very mall value for t < 0:5 mm. Therefore generally the contribution of t hould be ignored for the very thin ample and the G factor i obtained again a Equation (30) Œ Cylindrical geometry lim K D R R!0 4 : (45) Equation (46) i applicable for very high ditance according to radiu ( R/ and reitivity i obtained a Equation (47) Œ9. Again here, in order to obtain Equation (47), R hould go to zero o the ample hould be very thin. Thi expreion belong to a cylinder that ha finitely thin thickne. lim K D R R! ; (46) D V R D V G: (47) Yamahita calculated the correction factor from Poion equation for the hollow conducting cylinder and depend on the extenion of the ample and arrangement of probe Œ0. However the computing of the correction factor which i obtained by Yamahita (987, 988) calculating from the Poion equation for the circular dik pellet i converged to and /ln Œ9; Concluion Fundamental geometric factor of and /ln are alway valid when the extenion are negligible. Special cae of

8 different correction factor have alway ued thee two coefficient. A hown above, there are many equation and mathematic method that have been obtained by different reearcher and all of them give the ame reult. Auming a correct calculation of reitivity and a ample geometry a given above (or uing an equation cloer to our decribed ample), the correction factor i correct with a mall error. The expreion for the commonly ued geometrie are preented in a table for handy reference (Table 5). Reference [] Yamahita M. Reitivity correction factor for the four-probe method. J Phy E: Sci ntrum, 987, 0: 454 [] Topoe H. Geometric factor in four-point reitivity meaurement. Semiconductor Diviion No (Vedbaek), 968: 38 [3] Radiometer analytical, conductivity theory and practice. Radiometer Analytical D6M00, France, 003: 5 [4] Richard J A, Sear F W, Wehr M R, et al. Modern univerity phyic. t ed. London: Addion-Weley Publihing, 964: 4, 443 [5] Sze S M. Phyic of emiconductor device. John Wiley and Son, 969: 4 [6] Hee E. Reitivity meaurement of thin doped emiconductor layer by mean of four point-contact arbitrarily paced on a circumference of arbitrary radiu. Solid-State Electron,978, : 637 [7] Huang R S, Ladbrooke P H. The ue of a four-point probe for profiling ub-micron layer. Solid-State Electron, 978, : 3 [8] Valde L B. Reitivity meaurement on germanium for tranitor. Proce RE, 954, 4: 40 [9] Yamahita M. Reitivity correction factor for four-probe method on circular emiconductor. Jpn J Appl Phy, 988, 7: 37 [0] Albert M P. Correction factor for radial reitivity gradient evaluation of emiconductor lice. EEE Tran Electron Device, 964, : 48 [] Smit F M. Meaurement of heet reitivitie with the four-point probe. The Bell Sytem Technical Journal, 958, 37: 7 [] Swartzendruber L J. Correction factor table for four-point probe reitivity meaurement on thin, circular emiconductor ample. National Bureau of Standart Technical Note 99, Wahington, 964: [3] Swartzendruber L J. Four-point probe meaurement of nonuniformitie in emiconductor heet reitivity. Solid State onic, 964, 7: 43 [4] Vaughan D E. Four-probe reitivity meaurement on mall circular pecimen. Britih Journal of Applied Phyic, 96, : 44 [5] Yamahita M. Reitivity correction factor for four-probe method on circular emiconductor. Jpn J Appl Phy, 987, 6: 550 [6] Uhlir A. The potential of infinite ytem of ource and numerical olution of problem in emiconductor engineering. The Bell Sytem Technical Journal, 955, 34(): 05 [7] Yilmaz S. Synthei, characterization and invetigation of the olid tate oxygen ionic conductivitie of beta-bi O 3 type olid electrolyte doped with Dy O 3, Eu O 3, Sm O 3. PhD Thei, Gazi Univerity, 008: 50 [8] Yamahita M, Enjoji H. Reitivity correction factor fort the fourcircular-probe method. Jpn J Appl Phy, 989, 8(): 58 [9] Gegenwarth H H. Correction factor for the four-point probe reitivity meaurement on cylincdrical emiconductor. Solid- State Electron, 968, : 787 [0] Yamahita M. Meauring reitivity of hollow conducting cylindire with a four-probe array. Meaurement Science and Technology, 006, 7: